Debye screening versus Gauss law in electrostatics: Finite size effects

We revisit the well-known topics of self- and induced-screening in an otherwise isotropic neutral plasma/colloid. It is pointed out that the standard Debye–Hückel (DH) theory (ignoring finite size effects) suffers from many ambiguities related to net ionic numbers, total charge of the system, role of the electrostatic Gauss law, short-distance behaviour of the potential and incorrectly normalized pair correlation functions. We give a new formulation (incorporating finite size effects) such that ionic numbers are maintained, the total charge of the system has physically correct value, the Gauss law boundary conditions are rigorously obeyed, short-distance behaviour of the potential is guaranteed automatically, and correlation functions are correctly normalized. Numerical differences between the two approaches show up if the screening length μ−1 becomes comparable to the size R of the system.